A proof of the statement has already been given, so I will just add a small historical remark.
The left hand side of your identity is the <a href="https://en.wikipedia.org/wiki/Veneziano_amplitude">Veneziano amplitude</a> in the case of four identical scalar particles. The right hand side corresponds to another crossing symmetric four-point amplitude for scalar particles suggested by Virasoro in <a href="https://doi.org/10.1103/PhysRev.177.2309">"Alternative Constructions of Crossing-Symmetric Amplitudes with Regge Behavior"</a>. These are known to coincide for the choice of parameters $a+b+c=1$ giving your identity. I would recommend the article <a href="https://arxiv.org/abs/0704.0101">"The birth of string theory"</a> for a quick survey of the Veneziano model and its generalization to N scalar particles and related mathematics.